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We present a new and relatively elementary method for studying the solutionof the initial-value problem for dispersive linear and integrable equations inthe large-$t$ limit, based on a generalization of steepest descent techniquesfor Riemann-Hilbert problems to the setting of $\overline{\partial}$-problems.Expanding upon prior work of the first two authors, we develop the method indetail for the linear and defocusing nonlinear Schr\"odinger equations, andshow how in the case of the latter it gives sharper asymptotics than previouslyknown under essentially minimal regularity assumptions on initial data.

Dispersive Asymptotics for Linear and Integrable Equations by the $\overline{\partial}$ Steepest Descent Method